Question 1
Probability calculator
Distribution: Log normal
Mean log : 0
St. dev log : 1
Lower bound : 0
Upper bound : 1
P(X < 0) = 0
P(X > 0) = 1
P(X < 1) = 0.5
P(X > 1) = 0.5
P(0 < X < 1) = 0.5
1 - P(0 < X < 1) = 0.5
What is the probability that a log-normal random variable [in R, lnorm is the noun] whose logarithm has mean equal to 0 and standard deviation equal to 1, takes values less than 1?
0.5
Question 2
Probability calculator
Distribution: Log normal
Mean log : 0
St. dev log : 1
Lower bound : 0
Upper bound : 0.95
P(X < 0) = 0
P(X > 0) = 1
P(X < 5.18) = 0.95
P(X > 5.18) = 0.05
P(0 < X < 5.18) = 0.95
1 - P(0 < X < 5.18) = 0.05
What is the value of a log-normal random variable [in R, lnorm is the noun] whose logarithm has mean equal to 0 and standard deviation equal to 1, such that 0.95 of the probability is below said value?
5.18
Question 3
Plot a random sample of 1000 random draws from the aforementioned lognormal distribution (mean 0 and std. dev. 1).
The purpose is to have you visualize the distribution as a sanity check for 1 and 2.
Question 4
Probability calculator
Distribution: Normal
Mean : 0
St. dev : 1
Lower bound : 0.15865
Upper bound : 0.84135
P(X < -1) = 0.15865
P(X > -1) = 0.841
P(X < 1) = 0.84135
P(X > 1) = 0.159
P(-1 < X < 1) = 0.6827
1 - P(-1 < X < 1) = 0.317
The central probability in a standard normal distribution between -1 and 1 is 0.6827.
** Figure out z such that the probability between XXX and XXX is 0.6827. The middle of a z (normal 0, 1) is 0 so the probability below 0 is the same as the probability above zero which is 0.5. Now we need 0.6827 to each side. First, split it into two parts. There will be 0.6827/2 on each side of zero. So the first value must then be the solution to 0.5 - 0.34135 or 0.15865: -1. The second value will be 0.5 + 0.34135 or 0.84135: 1.**
Newspapers
Two key features are of note:
- The average price is clearly lower at Henry Hub
- The amount of variation at Henry Hub is considerably larger than at Tianjin in spite of the fact that the average is lower.
Newspapers: Summary
| Name | Newspapers |
| Number of rows | 61 |
| Number of columns | 7 |
| _______________________ | |
| Column type frequency: | |
| factor | 1 |
| numeric | 3 |
| POSIXct | 3 |
| ________________________ | |
| Group variables | None |
Variable type: factor
| skim_variable | n_missing | complete_rate | ordered | n_unique | top_counts |
|---|---|---|---|---|---|
| Location | 0 | 1 | FALSE | 2 | Hen: 36, Tia: 25 |
Variable type: numeric
| skim_variable | n_missing | complete_rate | mean | sd | p0 | p25 | p50 | p75 | p100 | hist |
|---|---|---|---|---|---|---|---|---|---|---|
| Price | 0 | 1.00 | 3.88 | 0.15 | 3.52 | 3.77 | 3.92 | 4.00 | 4.07 | ▂▃▃▅▇ |
| HHPrice | 25 | 0.59 | 3.80 | 0.14 | 3.52 | 3.69 | 3.80 | 3.90 | 4.07 | ▃▆▇▆▃ |
| TianjinPrice | 36 | 0.41 | 3.99 | 0.05 | 3.88 | 3.98 | 4.00 | 4.04 | 4.05 | ▂▂▂▇▇ |
Variable type: POSIXct
| skim_variable | n_missing | complete_rate | min | max | median | n_unique |
|---|---|---|---|---|---|---|
| Date | 0 | 1.00 | 2013-05-04 | 2013-10-30 | 2013-08-06 00:00:00 | 61 |
| HHDate | 25 | 0.59 | 2013-05-04 | 2013-10-25 | 2013-07-29 12:00:00 | 36 |
| TianjinDate | 36 | 0.41 | 2013-05-05 | 2013-10-30 | 2013-08-13 00:00:00 | 25 |
Two key features are of note:
- The average price is clearly lower at Henry Hub
- The amount of variation at Henry Hub is considerably larger than at Tianjin in spite of the fact that the average is lower.
Experts
Almost everything that we wish to know can be discerned from this.
- Because we can see 100 on the y-axis, we can get a sense of the missing responses; there are far more for Tianjin than Henry Hub.
- Experts clearly favor Henry Hub; the split in Build/No is nearly even for Tianjin but overwhelmingly favors Build for Henry Hub.
Experts: Tables
The Raw Table
Location Build No Total
HenryHub 66 15 81
Tianjin 31 33 64
Total 97 48 145A Percentage Table
Location Build No
HenryHub 0.8148148 0.1851852
Tianjin 0.4843750 0.5156250- Henry Hub has far more responses and more favorable responses than Tianjin.
- Experts clearly favor Henry Hub; the split in Build/No is nearly even for Tianjin but overwhelmingly favors Build for Henry Hub.
Forecast
Two things are of note:
1. The data are paired by period; that is the point of the calibration exercise.
2. Henry Hub is almost always forecast higher than Tianjin. Choose a period and you can hover over the associated values to convince yourself of this. Compare the two forecasts for any given period; they are almost always higher for Henry Hub.
Forecast: Data
Henry Hub is almost always forecast higher than Tianjin. The difference column captures this perfectly and shows that it always favors Henry Hub.